Average Error: 31.8 → 0.3
Time: 15.0s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03169561986009639120709380222251638770103 \lor \neg \left(x \le 0.03427235887658296870084129182032484095544\right):\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03169561986009639120709380222251638770103 \lor \neg \left(x \le 0.03427235887658296870084129182032484095544\right):\\
\;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\end{array}
double f(double x) {
        double r18961 = 1.0;
        double r18962 = x;
        double r18963 = cos(r18962);
        double r18964 = r18961 - r18963;
        double r18965 = r18962 * r18962;
        double r18966 = r18964 / r18965;
        return r18966;
}
↓
double f(double x) {
        double r18967 = x;
        double r18968 = -0.03169561986009639;
        bool r18969 = r18967 <= r18968;
        double r18970 = 0.03427235887658297;
        bool r18971 = r18967 <= r18970;
        double r18972 = !r18971;
        bool r18973 = r18969 || r18972;
        double r18974 = 1.0;
        double r18975 = cos(r18967);
        double r18976 = r18974 - r18975;
        double r18977 = 1.0;
        double r18978 = r18977 / r18967;
        double r18979 = r18978 / r18967;
        double r18980 = r18976 * r18979;
        double r18981 = 0.001388888888888889;
        double r18982 = 4.0;
        double r18983 = pow(r18967, r18982);
        double r18984 = r18981 * r18983;
        double r18985 = 0.5;
        double r18986 = r18984 + r18985;
        double r18987 = 0.041666666666666664;
        double r18988 = 2.0;
        double r18989 = pow(r18967, r18988);
        double r18990 = r18987 * r18989;
        double r18991 = r18986 - r18990;
        double r18992 = r18973 ? r18980 : r18991;
        return r18992;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03169561986009639 or 0.03427235887658297 < x

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.2

      \[\leadsto \frac{\color{blue}{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}}{x \cdot x}\]

    4. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}}\]
    5. Using strategy rm

    6. Applied div-inv0.6

      \[\leadsto \frac{\sqrt{1 - \cos x}}{x} \cdot \color{blue}{\left(\sqrt{1 - \cos x} \cdot \frac{1}{x}\right)}\]

    7. Applied div-inv0.6

      \[\leadsto \color{blue}{\left(\sqrt{1 - \cos x} \cdot \frac{1}{x}\right)} \cdot \left(\sqrt{1 - \cos x} \cdot \frac{1}{x}\right)\]

    8. Applied swap-sqr0.6

      \[\leadsto \color{blue}{\left(\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right)}\]

    9. Simplified0.6

      \[\leadsto \color{blue}{\left(1 - \cos x\right)} \cdot \left(\frac{1}{x} \cdot \frac{1}{x}\right)\]

    10. Simplified0.5

      \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{\frac{1}{x}}{x}}\]

    if -0.03169561986009639 < x < 0.03427235887658297

    1. Initial program 62.2

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03169561986009639120709380222251638770103 \lor \neg \left(x \le 0.03427235887658296870084129182032484095544\right):\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))